Theorem onintexmid 4695

Description: If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)

Hypothesis

Ref	Expression
onintexmid.onint	⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦)

Assertion

Ref	Expression
onintexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem onintexmid

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prssi 3852	. . . . . 6 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → {𝑢, 𝑣} ⊆ On)
2		prmg 3814	. . . . . . 7 ⊢ (𝑢 ∈ On → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
3	2	adantr 276	. . . . . 6 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
4		zfpair2 4323	. . . . . . 7 ⊢ {𝑢, 𝑣} ∈ V
5		sseq1 3261	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → (𝑦 ⊆ On ↔ {𝑢, 𝑣} ⊆ On))
6		eleq2 2296	. . . . . . . . . 10 ⊢ (𝑦 = {𝑢, 𝑣} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ {𝑢, 𝑣}))
7	6	exbidv 1874	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → (∃𝑥 𝑥 ∈ 𝑦 ↔ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}))
8	5, 7	anbi12d 473	. . . . . . . 8 ⊢ (𝑦 = {𝑢, 𝑣} → ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) ↔ ({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣})))
9		inteq 3952	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → ∩ 𝑦 = ∩ {𝑢, 𝑣})
10		id 19	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → 𝑦 = {𝑢, 𝑣})
11	9, 10	eleq12d 2303	. . . . . . . 8 ⊢ (𝑦 = {𝑢, 𝑣} → (∩ 𝑦 ∈ 𝑦 ↔ ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣}))
12	8, 11	imbi12d 234	. . . . . . 7 ⊢ (𝑦 = {𝑢, 𝑣} → (((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦) ↔ (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})))
13		onintexmid.onint	. . . . . . 7 ⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦)
14	4, 12, 13	vtocl 2869	. . . . . 6 ⊢ (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})
15	1, 3, 14	syl2anc 411	. . . . 5 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})
16		elpri 3712	. . . . 5 ⊢ (∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣} → (∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣))
17	15, 16	syl 14	. . . 4 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → (∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣))
18		incom 3411	. . . . . . 7 ⊢ (𝑣 ∩ 𝑢) = (𝑢 ∩ 𝑣)
19	18	eqeq1i 2240	. . . . . 6 ⊢ ((𝑣 ∩ 𝑢) = 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑢)
20		dfss1 3425	. . . . . 6 ⊢ (𝑢 ⊆ 𝑣 ↔ (𝑣 ∩ 𝑢) = 𝑢)
21		vex 2816	. . . . . . . 8 ⊢ 𝑢 ∈ V
22		vex 2816	. . . . . . . 8 ⊢ 𝑣 ∈ V
23	21, 22	intpr 3981	. . . . . . 7 ⊢ ∩ {𝑢, 𝑣} = (𝑢 ∩ 𝑣)
24	23	eqeq1i 2240	. . . . . 6 ⊢ (∩ {𝑢, 𝑣} = 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑢)
25	19, 20, 24	3bitr4ri 213	. . . . 5 ⊢ (∩ {𝑢, 𝑣} = 𝑢 ↔ 𝑢 ⊆ 𝑣)
26	23	eqeq1i 2240	. . . . . 6 ⊢ (∩ {𝑢, 𝑣} = 𝑣 ↔ (𝑢 ∩ 𝑣) = 𝑣)
27		dfss1 3425	. . . . . 6 ⊢ (𝑣 ⊆ 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑣)
28	26, 27	bitr4i 187	. . . . 5 ⊢ (∩ {𝑢, 𝑣} = 𝑣 ↔ 𝑣 ⊆ 𝑢)
29	25, 28	orbi12i 772	. . . 4 ⊢ ((∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣) ↔ (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
30	17, 29	sylib 122	. . 3 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
31	30	rgen2a 2596	. 2 ⊢ ∀𝑢 ∈ On ∀𝑣 ∈ On (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)
32	31	ordtri2or2exmid 4693	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2203   ∩ cin 3210   ⊆ wss 3211  {cpr 3690  ∩ cint 3949  Oncon0 4484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-int 3950  df-tr 4209  df-iord 4487  df-on 4489  df-suc 4492

This theorem is referenced by: (None)